On De Giorgi’s conjecture in dimensions 4 and 5

Abstract

In this paper, we develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago. The problem originates in theory of phase transition and is so closely connected to theory of minimal hypersurfaces that it is sometimes referred to as the eversion of Bernstein's problem for minimal graphs. The conjecture has been completely settled in dimension 2 by authors [15] and in dimension 3 in [2], yet approach in this paper seems to be first to use, in an essential way, solution of Bernstein problem stating that minimal graphs in Euclidean space are necessarily hyperplanes provided dimension of ambient space is not greater than 8. We note that solution of Bernstein's problem was also used in [18] to simplify an argument in [9]. Here is conjecture as stated by De Giorgi [12]. CONJECTURE 1.1. Suppose that u is an entire solution of equation